Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System

0.38 MB·English

by  Aristophanes Dimakis

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5(2009), 002, 18 pages
Multicomponent Burgers and KP Hierarchies,
⋆
and Solutions from a Matrix Linear System
Aristophanes DIMAKIS † and Folkert MU¨LLER-HOISSEN ‡
† Department of Financial and Management Engineering,
University of the Aegean, 31 Fostini Str., GR-82100 Chios, Greece
E-mail: [email protected]
‡ Max-Planck-Institute for Dynamics and Self-Organization,
Bunsenstrasse 10, D-37073 Go¨ttingen, Germany
9
E-mail: [email protected]
0
0 Received November 01, 2008, in final form January 04, 2009; Published online January 08, 2009
2
doi:10.3842/SIGMA.2009.002

n
a
J Abstract. ViaaCole–Hopftransformation,themulticomponentlinearheathierarchyleads

8 toamulticomponentBurgershierarchy. Weshowinparticularthatanysolutionofthelatter
also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of
]
I theCole–Hopftransformationleadstoamoregeneralrelationbetweenthe multicomponent
S linearheathierarchyandthemulticomponentKPhierarchy. Fromthisresultsaconstruction
.
n of exact solutions of the latter via a matrix linear system.
i
l Key words: multicomponent KP hierarchy; Burgers hierarchy; Cole–Hopf transformation;
n
[ Davey–Stewartsonequation; Riccati equation; dromion

2 2000 Mathematics Subject Classification: 37K10;35Q53
v
0
1
1 1 Introduction
0
.
1
Thewell-known Cole–Hopf transformationφ= ψ ψ−1 translates thenonlinearBurgersequation
1 x
8 into the linear heat equation [1, 2, 3]. This extends to a corresponding relation between the
0
Burgers hierarchy and the linear heat hierarchy, and moreover to their matrix generalizations.
:
v The Cole–Hopf transformation also generates solutions of the KP hierarchy from (invertible)
i
X solutions of the linear heat hierarchy [4, 5, 6], and this extends to the corresponding matrix
r hierarchies[7,8,9]. Thereisalsoageneralization oftheCole–Hopftransformationthatproduces
a
a solution of the (scalar or matrix) KP hierarchy from two solutions of the matrix linear heat
hierarchy, connected by an additional relation [7, 8]. Furthermore, this includes a construction
of (matrix) KP solutions from a matrix linear system (see also [10, 11]), which may be regarded
as a finite-dimensional version of the Sato theory [12, 13]. In this work we extend these results
to the multicomponent case. Admittedly, this is not a very difficult task, on the basis of our
previous results [7, 11]. We also take this opportunity, however, to present these results in
a concise form and to add some relevant remarks and examples.
The usual multicomponent KP (mcKP) hierarchy contains subhierarchies that generalize
the matrix KP hierarchy by modifying it with some constant matrix B different from the
identity matrix. We are particularly interested in such matrix hierarchies different from the
ordinary matrix KP hierarchy. The Davey–Stewartson equation [14, 15], a two-dimensional
nonlinear Schro¨dinger equation that appeared as a shallow-water limit of the Benney–Roskes
⋆This paper is a contribution to the Proceedings of the XVIIth International Colloquium on Integrable Sys-
tems and Quantum Symmetries (June 19–22, 2008, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2008.html
2 A. Dimakis and F. Mu¨ller-Hoissen
equation [16, 17], emerges from such a modified matrix hierarchy (see Section 3). It is of course
well-known to arise from the two-component KP hierarchy.
Section 2 recalls the Cole–Hopf transformation for a multicomponent Burgers (mcBurgers)
hierarchy and Section 3 reveals relations between the mcBurgers and the mcKP hierarchy. Sec-
tion4presentstheabovementionedkindofgeneralizationoftheCole–Hopftransformationwhich
determines solutions of the mcKP hierarchy. From this derives a fairly simple method to con-
struct mcKP solutions. This is the subject of Section 5, which also presents some examples.
All this is closely related to a multicomponent version of a matrix Riccati hierarchy, as briefly
explained in Section 6. Section 7 contains some conclusions.
In the following, let be an associative (and typically noncommutative) algebra over the
A
complex (or real) numbers, with identity element I and supplied with a structure that allows
to define derivatives with respect to real variables. Let be a finite set of mutually commuting
B
elementsof . Althoughthisassumptionissufficienttoestablishtheresultsinthiswork,further
A
assumptionsshouldbeplaced on inparticularinordertodiminishredundancy,seetheremark
B
at the end of Section 2. With each B we associate a sequence of (real or complex) variables
∈ B
t = (t ,...,t ,...). Furthermore, for a function f of t we define a Miwa shift with
B B,1 B,n B B∈B
{ }
respect to B by f (...,t ,...)= f(...,t +[λ],...), where [λ] = (λ,λ/2,λ/3,...) and λ
∈ B [λ]B B B
is an indeterminate. f denotes the partial derivative of f with respect to the variable t .
tn n
2 Cole–Hopf transformation
for a multicomponent Burgers hierarchy
Let us consider the multicomponent linear heat hierarchy
ψ = Bn∂n(ψ) B , n = 1,2,..., (2.1)
tB,n ∀ ∈B
where ∂ = ∂ is the operator of partial differentiation with respect to a variable x, and ψ has
x
values in . Since ψ = Bψ , this implies the ordinary linear heat hierarchy ψ = ∂n (ψ).
A tB,1 x tB,n tB,1
Any two flows (2.1) commute as a consequence of our assumptions for (the elements do not
B
depend on t and commute with each other). A functional representation1 of (2.1) is
B B∈B
{ }
given by
λ−1(ψ ψ ) = Bψ B . (2.2)
− −[λ]B x ∀ ∈ B
Proposition 1. If ψ is an invertible solution of the above multicomponent linear heat hierarchy,
then
φ = ψ ψ−1 (2.3)
x
solves the multicomponent Burgers (mcBurgers) hierarchy associated with , given by the func-
B
tional representation
Ω (φ,λ) = 0 B , (2.4)
B
∀ ∈ B
where
Ω (φ,λ) := λ−1(φ φ ) (Bφ φ B)φ Bφ . (2.5)
B − −[λ]B − − −[λ]B − x
1Weusethis(notquitesatisfactory) termforanequationthatgeneratesasequenceofequationsbyexpansion
in powers of an indeterminate.
Multicomponent Burgers and KP Hierarchies 3
Proof. We have to consider the following system,
ψ = φψ, ψ = (I λBφ)ψ B .
x −[λ]B − ∀ ∈ B
The integrability condition (ψ ) = (ψ ) yields (2.4). The further integrability condi-
x −[λ]B −[λ]B x
tion (ψ ) = (ψ ) is
−[λ]B1 −[µ]B2 −[µ]B2 −[λ]B1
B λ−1(φ φ )+φ B φ = B µ−1(φ φ )+φ B φ ,
2 − −[λ]B1 −[λ]B1 1 1 − −[µ]B2 −[µ]B2 2
(cid:0) (cid:1) (cid:0) (cid:1)
which is satisfied as a consequence of (2.4) and [B ,B ] = 0. (cid:4)
1 2
(2.3) is a Cole–Hopf transformation. The first equation that results from (2.4) is
φ = Bφ +[B,φ]φ,
tB,1 x
which has been called C-integrable N-wave equation in [9]. If B = I, this reduces to t = x.
I,1
But it is a nontrivial nonlinear equation if φ does not commute with B. The next equation that
results from (2.4) is the (noncommutative) Burgers equation
φ = φ +2φ Bφ.
tB,2 tB,1tB,1 tB,1
Remark 1. In order to avoid redundancy and to maximally extend the hierarchies, further
conditions have to beimposed on . In particular, theelements of should belinearly indepen-
B B
dent, since linear combinations correspond to linear combinations of hierarchy equations. But
since products of elements of also generate (independent or redundant) commuting flows, the
B
problem is more subtle. If the algebra is semisimple, it admits a maximal set of commuting mu-
tually annihilating idempotents E , a = 1,...,N, hence E E = δ E for all a,b = 1,...,N.
a a b a,b a
Then = E N is an optimal choice. In fact, in the following we do not really address those
B { a}k=1
cases of (non-semisimple) algebras where such a choice does not exist. Rather it turns out that
some more flexibility in the choice of can be used to obtain certain integrable systems within
B
this framework in a more direct way, see Section 3.
3 Multicomponent KP and relations with the multicomponent
Burgers hierarchy
For B let
∈ B
(λ) := I λ(ω (λ)+B∂).
B B
E −
The “discrete” zero curvature condition
(λ) (µ)= (µ) (λ) (3.1)
EB1 −[µ]B2EB2 EB2 −[λ]B1EB1
then leads to the two equations
λ−1(ω (µ) ω (µ) )+ω (µ) ω (λ)+B ω (λ)
B2 − B2 −[λ]B1 B2 −[λ]B1 B1 2 B1 x
= µ−1(ω (λ) ω (λ) )+ω (λ) ω (µ)+B ω (µ)
B1 − B1 −[µ]B2 B1 −[µ]B2 B2 1 B2 x
and
B ω (λ) ω (λ) B = B ω (µ) ω (µ) B .
2 B1 − B1 −[µ]B2 2 1 B2 − B2 −[λ]B1 1
The last equation is solved by
ω (λ) = Bφ φ B,
B − −[λ]B
4 A. Dimakis and F. Mu¨ller-Hoissen
and the first equation can then be written in terms of (2.5) as
B Ω (φ,λ) Ω (φ,λ) B = B Ω (φ,µ) Ω (φ,µ) B B ,B .(3.2)
2 B1 − B1 −[µ]B2 2 1 B2 − B2 −[λ]B1 1 ∀ 1 2∈B
We take this as our defining equations of the (more precisely, potential) multicomponent KP
(mcKP) hierarchy associated with .2
B
Remark 2. Choosing B = B = B in (3.2) and summing the resulting equation three times
1 2
with cyclically permuted indeterminates, leads to
3
ǫ λ−1(φ φ )+φ Bφ B = 0,
ijk i − −[λi]B −[λi]B −[λj]B
i,jX,k=1 (cid:0) (cid:1)
which is a special case of the functional form of the mcKP hierarchy in [27].
Let us take a closer look at (3.2) with B = B = B. Its λ-independent part is
1 2
BΩ (φ,0) Ω (φ,0) B = [B,Ω (φ,µ)],
B − B −[µ]B B
where Ω (φ,0) = φ [B,φ]φ Bφ . To first order in µ this gives
B tB,1 − − x
Bφ B 1 B,φ + 1[B,φ ]= Bφ [B,φ] [B,φ]φ B. (3.3)
tB,1x − 2{ tB,1tB,1} 2 tB,2 tB,1 − tB,1
(3.2) is the integrability condition of
Ω (φ,λ) = Bθ θ B B , (3.4)
B − −[λ]B ∀ ∈B
withanewdependentvariableθ. (3.4)representsthemcKPhierarchyintermsoftwo dependent
variables. Bycomparisonwith(2.4),thishastheformofaninhomogeneous mcBurgershierarchy.
The following is an immediate consequence3.
Proposition 2. Any solution of the mcBurgers hierarchy also solves the mcKP hierarchy.
Proof. (3.4) becomes (2.4) if θ = 0. (cid:4)
The representation (3.4) of the mcKP hierarchy has the advantage that each equation only
involves a single element from the set . To order λ0, (3.4) yields
B
φ Bφ [B,φ]φ = [B,θ]. (3.5)
tB,1 − x−
If B = I, then (3.5) reduces to φ = φ . Otherwise this is a non-trivial nonlinear equation.
tB,1 x
Subtracting (3.5) from (3.4), leads to
λ−1(φ φ ) φ (φ φ )Bφ= (θ θ )B. (3.6)
− −[λ]B − tB,1 − − −[λ]B − −[λ]B
The two equations (3.5) and (3.6) are equivalent to (3.4). (3.6) does not involve derivatives with
respect to x. For fixed B , it represents the KP hierarchy in [7], with product modified
∈ B A
by B.
2Seealso e.g. [18,19,20,21,22,6,23,24,25,26,27]fordifferent formulations ofsuchamulticomponent KP
hierarchy. We should also mention that the conditions imposed on the set B can be relaxed while keeping the
hierarchy property,see [26] for example.
3Itwasfirstnotedin[4]thatanysolutionofthefirsttwoequationsofthe(scalar)Burgershierarchyalsosolves
the (scalar potential) KP equation. In [28, 29] the (first two) Burgers hierarchy equations have been recovered
via a symmetry constraint from theKPhierarchy and its linear system.
Multicomponent Burgers and KP Hierarchies 5
Proposition 3. For any B , as a consequence of the mcKP hierarchy (3.4), Bφ and also
∈ B
φB solves the ordinary (noncommutative) KP hierarchy.
Proof. This is an immediate consequence of (3.6). (cid:4)
To first order in λ−1, (3.6) yields
1(φ φ ) φ Bφ=θ B.
2 tB,2 − tB,1tB,1 − tB,1 tB,1
Differentiating (3.5) with respect to t , and multiplying it by B from the right, we can use
B,1
the last equation to eliminate θ. In this way we recover (3.3). As shown in Example 2 below,
(3.3) generalizes the Davey–Stewartson (DS) equation [16, 14, 15, 17]. Eliminating θ from (3.6)
by use of (3.5) thus leads to a (generalized) DS hierarchy.
Example 1. Let B = σ, where σ2 = I. Decomposing φ as
φ = D+U, where D := 1(φ+σφσ), U := 1(φ σφσ),
2 2 −
(3.3) splits into the two equations4
D σ D 2U2 = 0, (3.7)
t1 − x −
U σ+U +2 U,D =0, (3.8)
t2 xt1 { t1}
wherewe write t , t instead of t and t . Let now bethe algebra of 2 2 matrices over C,
1 2 B,1 B,2
A ×
and σ = diag(1, 1). Then D and U are diagonal and off-diagonal parts of φ, respectively.
−
Writing
0 u
U = , hence u := φ , v := φ , (3.9)
(cid:18) v 0 (cid:19) 1,2 2,1
and introducing
s := tr(φ) = φ +φ , r := φ φ ,
1,1 2,2 1,1 2,2
−
we obtain the system
u u 2us = 0, v +v +2vs = 0, (3.10)
t2 − xt1 − t1 t2 xt1 t1
and s = r , s = r 4uv. The integrability conditions of the latter two equations are
t1 x x t1 −
s s = 4(uv) , r r = 4(uv) . (3.11)
t1t1 − xx x t1t1 − xx t1
The two equations (3.10) together with the first of (3.11) constitute a fairly simple system
of three nonlinear coupled equations, where all variables can be taken to be real. Allowing
complex dependent and independent variables, after a complex transformation the system for
thedependentvariablesu, v, scan befurtherreduced tothe DS equation, see thenext example.
Of course, the above system (3.10) and(3.11) can also bederived from theusualtwo-component
KP hierarchy (see e.g. [27]), and the transformation to DS is well-known.
Setting v = 0, we obtain from (3.10) and (3.11) the following linear equations5,
u u 2ϕu = 0, ϕ ϕ = 0, (3.12)
t2 − xt1 − t1t1 − xx
whereϕ := s = tr(φ) . Thisisprobablythesimplestsystem thatpossessesdromion solutions,
t1 t1
as observed in Example 3 in Section 5.
4Aconstant(withrespecttot )ofintegrationhasbeensettozeroinordertoobtain(3.7). Thelatterequation
1
can beobtained more directly as the diagonal part of (3.5).
5If u does not depend on t , then the first equation is part of a Lax pair for the Nizhnik–Novikov–Veselov
2
equation.
6 A. Dimakis and F. Mu¨ller-Hoissen
Example 2. We continue with the previous example and perform a transformation to the DS
equation by first allowing the dependent variables to live in a noncommutative algebra. In this
way we obtain a certain noncommutative generalization of the DS equation (see also [30, 31] for
matrix DS versions). In terms of
F = D +βU2,
t1
with β C, (3.7) becomes
∈
F σ F = β(U2) +(U2) (2+βσ).
t1 − x − x t1
Differentiating this with respect to x and with respectto t , respectively, and eliminating mixed
1
derivatives of F from the resulting two equations, yields
F F = β(U2) +2(U2) +(U2) (2σ +β). (3.13)
t1t1 − xx − xx xt1 t1t1
Furthermore, (3.8) takes the form
U σ+U = 2 U,F +4βU3. (3.14)
t2 xt1 − { }
Letnow bethealgebraof2 2matricesover someunitalassociative algebrawithunitdenoted
A ×
by 1, and σ = diag(1, 1). Using (3.9) and writing
−
F = diag(f,g),
(3.13) and (3.14) result in the following equations,
u u = 2(fu+ug) 4βuvu,
t2 − xt1 −
v +v = 2(gv+vf)+4βvuv,
t2 xt1 −
f f = β(uv) +2(uv) +(2+β)(uv) ,
t1t1 − xx − xx xt1 t1t1
g g = β(vu) +2(vu) (2 β)(vu) .
t1t1 − xx − xx xt1 − − t1t1
In terms of the new variables
1+i 1 i
y = (x+it ), z = − (x it ), (3.15)
1 1
κ√2 √2 −
with a constant κ= 0, and with the choice β = i, this becomes
6
1
u +u + u = 2(fu+ug) 4iuvu,
t2 zz κ2 yy −
1
v +v + v = 2(gv+vf)+4ivuv,
− t2 zz κ2 yy
1 1 2i
f f = (1+2i)(uv) (uv) (uv) ,
zz − κ2 yy zz − κ2 yy − κ yz
1 1 2i
g g = ( 1+2i)(vu) + (vu) + (vu) . (3.16)
zz − κ2 yy − zz κ2 yy κ yz
If the dependent variables take their values in a commutative algebra, then we obtain
1
u +u + u = 2ρu 4iu2v,
t2 zz κ2 yy −
1
v +v + v = 2ρv+4iv2u,
− t2 zz κ2 yy
1
ρ ρ = 4i(uv) ,
zz − κ2 yy zz
Multicomponent Burgers and KP Hierarchies 7
where
ρ := f +g = ϕ+2iuv (3.17)
(with ϕ= tr(φ) ). Setting t = it and
t1 2 −
v = iǫu¯ with ǫ = 1
±
and the complex conjugate u¯ of u, one recovers the DS system
1 1
iu +u + u = 2ρu+4ǫ u2u, ρ ρ = 4ǫ(u2) , (3.18)
t zz κ2 yy | | zz − κ2 yy − | | zz
where now y, z, t, κ2, ρ are taken to be real. Clearly, this system is more quickly obtained
from (3.10) and (3.11) by application of the transformation of independent variables given
by (3.15) and t = it, and the transformation (3.17) of dependent variables. But on our way
2
−
we obtained the system (3.16) which (with t = it) may be of interest as a noncommutative
2
−
version of the DS equation (see [30, 31] for alternatives).
We conclude that the DS equation (and a correspondinghierarchy) is obtained from a matrix
KP hierarchy, but the latter has to be generalized by introduction of a matrix B different from
theunitmatrix. Ofcourse,thishierarchyisembeddedintheusualtwo-componentKPhierarchy.
3.1 The associated Sato–Wilson system and its translation
into a Burgers hierarchy
(3.1) is the integrability condition of the linear system
ψ˜ = (λ)ψ˜ B .
−[λ]B EB ∀ ∈ B
If W is an invertible solution of
(λ)W = W (λ), (λ) := I λB∂, (3.19)
EB −[λ]BE0,B E0,B −
then the linear system is mapped to ψ = (λ)ψ where ψ := W−1ψ˜. The latter is the
−[λ]B E0,B
linear heat hierarchy (2.2). The ansatz
W = I + w ∂−n,
n
nX≥1
inserted into the functional form (3.19) of the Sato–Wilson equations, leads to
λ−1(w w ) Bw (Bφ φ B)w = Bw w B, (3.20)
n− n,−[λ]B − n,x− − −[λ]B n n+1− n+1,−[λ]B
wheren = 0,1,... andw = I. From then = 0 equation weget w = φ. In terms of θ := w ,
0 1 2
− −
the n = 1 equation turns out to be the functional form (3.4) of the mcKP hierarchy. Let us
introduce
Φ := Λ⊺+e (φ, w , w ,...), (3.21)
1 2 3
− −
⊺
where e = (I,0,...), and
1
0 I 0 0
···
Λ :=  0 0 I 0  (3.22)
···
 ... ... ... ... ... 
 
8 A. Dimakis and F. Mu¨ller-Hoissen
is the shift operator matrix with transpose Λ⊺. Then (3.20) can be expressed as the following
mcBurgers hierarchy,
Ω (Φ,λ) λ−1(Φ Φ ) (BΦ Φ B)Φ BΦ = 0 B . (3.23)
B ≡ − −[λ]B − − −[λ]B − x ∀ ∈ B
Thelinear system of the mcKP hierarchy has thus been reformulated as an mcBurgers hierarchy
with an infinite matrix variable of a special form.
We note that (3.21) has the form of a companion matrix, a block of a Frobenius normal
form matrix. This makes contact with recent work in [9], where several integrable systems were
recovered from equations of a Burgers hierarchy with dependent variable of Frobenius normal
form. All these systems are known to arise as reductions of the mcKP hierarchy.
⊺
As a consequence of (3.23) with Φ of the form (3.21), it follows that φ = e Φe solves the
1 1
mcKP hierarchy in .
A
In the following section, we shall see that Φ of the form (3.21) results, as a particular case,
from a quite general result about solutions of a somewhat generalized mcKP hierarchy.
4 Generalization of the Cole–Hopf transformation
We generalize (3.4) to
Ω (Φ,λ) = BΘ Θ B B , (4.1)
B,Q − −[λ]B ∀ ∈ B
where
Ω (Φ,λ) := λ−1(Φ Φ ) (BΦ Φ B)QΦ BΦ , (4.2)
B,Q − −[λ]B − − −[λ]B − x
withaconstantobjectQ. Weassumethattheconstituentsareelementsoflinearspacesandthat
the products are defined. (4.1) determines a generalization of the mcKP hierarchy, since non-
linear terms now involve Q (which modifies the product). The following generalizes a theorem
in [7] (see also [8]).
Theorem 1. Let X, Y be solutions of the multicomponent linear heat hierarchy, i.e.
λ−1(X X ) = BX , λ−1(Y Y ) = BY , (4.3)
− −[λ]B x − −[λ]B x
for all B , and furthermore
∈ B
X = RX +QY, (4.4)
x
with constant objects R, Q. If X is invertible and if all B commute with R, then
∈ B
Φ = YX−1 (4.5)
solves the mcKP hierarchy (4.1) with Θ = ΦR.
Q
Proof. Using (4.5) in the definition (4.2), we have
Ω (Φ,λ) = (BΦ Φ B)(X QY)X−1+(λ−1(Y Y ) BY )X−1
B,Q − −[λ]B x − − −[λ]B − x
Φ (λ−1(X X ) BX )X−1,
− −[λ]B − −[λ]B − x
which reduces to
Ω (Φ,λ) = (BΦ Φ B)R
B,Q − −[λ]B
as aconsequence of(4.3)and(4.4). Since[B,R]= 0, thistakes theform(4.1)withΘ = ΦR. (cid:4)
Multicomponent Burgers and KP Hierarchies 9
Nowwesetupthestageforapplications ofthetheorem. LetA(M,N) := Mat(M N,C) ,
× ⊗A
where Mat(M N,C) is the space of complex M N matrices. Let Φ, Θ, Y take values in
× ×
A(M,N), and X in A(N,N). Furthermore, let Q A(N,M) and R A(N,N) commute with
∈ ∈
all B .
∈ B
IfQhasrank one over ,inthesensethatQ = VU⊺withconstantvectorsU,V,withentries
A
in , and if U and V commute with all B , then φ = U⊺ΦV solves the mcKP hierarchy in ,
A ∈ B A
provided that Φ solves (4.1). In this way, any solution X, Y of the linear equations formulated
in the above theorem generates an -valued solution φ of the mcKP hierarchy (3.2).
A
Choosing M = N, (4.5) is a Cole–Hopf transformation if Y = X . Then (4.4) becomes
x
(I Q)X = RX. Let e denote the N-component vector with all entries zero except for the
x k
−
identity element in the kth row.
N−1
⊺ ⊺
a) Setting Q = e e and R = e e (which is the left shift operator: Re = e ,
N N k k+1 k k−1
kP=1
k = 2,...,N, and Re = 0), one finds that (4.4) restricts X to the form of a Wronski matrix
1
(see also [8]),
X(1) X(2) X(N)
···
 ∂(X(1)) ∂(X(2)) ∂(X(N)) 
X = .. .. ·.·.· .. .
 . . . . 
 
 ∂N−1(X(1)) ∂N−1(X(2)) ∂N−1(X(N)) 
 ··· 
X(1),X(2),...,X(N) areindependentfunctionswithvaluesin . Theremainingassumption(4.3)
A
in the theorem requires them to be solutions of the multicomponent heat hierarchy.
b) Let R = Λ⊺ with the infinite shift operator matrix (3.22), and Q = e e⊺. Then (4.4) says
1 1
that X has to be a pseudo-Wronski matrix
X(1) X(2) X(3)
···
 ∂−1X(1) ∂−1X(2) ∂−1X(3) 
X = ∂−2X(1) ∂−2X(2) ∂−2X(3) ··· , (4.6)
 ··· 
 ... ... ... ... 
 
where ∂−1 is the formal inverse of ∂. This structure appeared in [9] (equations (14) and (50)
therein). (4.3)demandsthatX(1),X(2),...solvethemulticomponentheathierarchy. With(4.6),
Φ = X X−1 has the form (3.21) and hence determines a solution of (3.23).
x
5 Solutions of the multicomponent KP hierarchy
from a matrix linear system
InordertoderivesomeclassesofmcKPsolutionsviatheorem1moreexplicitly, intheframework
specified in Section 4 (after the theorem) we extend (4.4) to
Z = HZ, (5.1)
x
where
X R Q
Z = , H = ,
(cid:18) Y (cid:19) (cid:18) S L (cid:19)
with new constant objects L A(M,M) and S A(M,N) that commute with all B . The
∈ ∈ ∈ B
two equations (4.3) then combine to
λ−1(Z Z ) =BZ B .
− −[λ]B x ∀ ∈ B
10 A. Dimakis and F. Mu¨ller-Hoissen
Taking (5.1) into account, this is equivalent to
Z = (BH)nZ B , n= 1,2,.... (5.2)
tB,n ∀ ∈ B
Note that B and H commute as a consequence of our assumptions. With a suitable choice of
the algebra , the solution of the matrix linear system (5.1), (5.2) is given by
A
Z(x,t) = eξ(x,t;H,B)Z , (5.3)
0
where
∞
ξ(x,t;H,B) := xH + t (HB)n,
B,n
BX∈BnX=1
and t stands for t . Decomposing Z into X and Y, theorem 1 implies that Φ = YX−1
B B∈B
{ }
solves the mcKP hierarchy (4.1). Furthermore, if rank(Q) = 1 over , hence Q = VU⊺ with
Q
A
constant vectors U and V, then the -valued variable
A
φ = U⊺ΦV
solves the corresponding mcKP hierarchy.
The exponential in (5.3) can be computed explicitly if additional assumptions are made
concerning the form of H (see [11], in particular). Then Φ is obtained via (4.5).
Case 1. Let S = 0 and
Q =RK KL (5.4)
−
with a constant N M matrix K (over ) that commutes with all B . Then we obtain
× A ∈ B
Φ = eξ(x,t;L,B)Ce−ξ(x,t;R,B) I Keξ(x,t;L,B)Ce−ξ(x,t;R,B) −1, (5.5)
N
−
(cid:0) (cid:1)
where I is the N N unit matrix over (so that the diagonal entries are the identity I
N
× A
in ), and C is an arbitrary constant M N matrix (with entries in ). Φ solves the mcKP
Q
A × A
hierarchy (associated with ), with Q given by (5.4).6 If moreover Q = VU⊺ with vectors U, V
B
that commute with all B , then φ = U⊺ΦV solves the mcKP hierarchy in . Of course, it
∈ B A
remains to solve the rank one condition (over )
A
RK KL = VU⊺. (5.6)
−
If M = N and if C is invertible, then (5.5) simplifies to
Φ = eξ(x,t;R,B)C−1e−ξ(x,t;L,B) K −1,
−
(cid:0) (cid:1)
which remains a solution if we replace C−1 by an arbitrary constant N N matrix C˜ (with
×
entries in ).
A
Example 3. Choosing the components of the matrices L, R as
L = p δ I, R = q δ I, (5.7)
ij i ij kl k kl
6In particular, if M = N and Q = IN, then Φ (with K, L, R solving RK−KL = IN) is a solution of the
N ×N matrix (over A) mcKP hierarchy.